Proof: We assume some basic facts about frame bundles, see  [1]},  [2]}. Given two Legendrian knots construct a smooth isotopy \(L_t\)  between them, this defines a path \(\beta _t: L \rightarrow V_{2n+1,n}\)  so that \(\beta _0\)  is a constant map (here \(V_{2n+1, n}\)  is the Stiefel manifold of \(n\) -frames in \(\mathbb {R}^{2n+1}\) ). \(L\)  need not admit a global parallelization, since here \(\beta _t\)  compares \(dL_t\)  to \(dL_0\)  at each point of \(L\) , and this difference does not depend on a choice of framing at that point. Said differently, maps \(L \rightarrow Gr_n(\mathbb {R}^{2n+1}_{std})\)  lifting the isotopy \(L_t\)  can be identified with \(\operatorname{Map}(L, V_{2n+1, n})\)  by choosing a connection on the tautological bundle over \(Gr_n(\mathbb {R}^{2n+1})\) . Inside \(V_{2n+1, n}\) , identify \(U_n\)  as the subset of Legendrian frames. (Though “which frames are Legendrian” depends on the point in \(\mathbb {R}^{2n+1}\) , these inclusions are all homotopy equvialent to the inclusion \(U_n \subseteq O_{2n} \subseteq O_{2n+1} \rightarrow V_{2n+1, n}\) .) \(\beta _1\)  has image inside of \(U_n\)  since \(L_1\)  is Legendrian, and so \(\beta _t\)  defines an element \(\beta \in \pi _1\left(\operatorname{Map}\left(L, V_{2n+1, n}\right), \operatorname{Map}\left(L, U_n\right)\right)\) . Notice that \(\operatorname{Map}(L, V_{2n+1, n})\)  is conected since \(V_{2n+1, n}\)  is \(n\) -connected.
